Eikonal Blog

2010.01.26

A Theorem of Ellis and Pinsky

Filed under: mathematics — Tags: — sandokan65 @ 13:12

Theorem (Ellis & Pinsky): Let $\latex A$, B be real symmetric n\times n matrices, and assume that B is negative semi-definite. Then as \epsilon \rightarrow +0:

  • e^{t(A+i\frac1{\epsilon}B)} e^{-i\frac{t}{\epsilon}B} = e^{t A_0} + {\cal O}(\epsilon),

where A_0 :\equiv \sum_{\lambda\in\sigma(B)} P_\lambda A P_\lambda, here P_\lambda being the orthogonal projector on the eigenspace of B associated with the eigenvalue \lambda.

Source: T1270 = S.L.Campbell “On the limit of a product of matrix exponentials”; Linear and Multilinear Algebra; 1978, Vol 6, p.p. 55-59.

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